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Internet
This category contains questions and answers about the World Wide Web and related subjects.
18,762 Questions
How do you change font text color sizes and style?
For your color you will just put in this code before your writing that you want a certain color.
Color Code- <"font color= "pink">
To end Color Code-
You can use any color you can think of.
For your sizes use- ==
What is the importance of ammeter?
An ammeter measures current, an important variable in electronics.
Who first used the term internet?
Jean Armor Polly is credited with coining the phrase, "surf the net". She said Internet instead of net, but it was shortened later by computer lingo geeks to net. Jean Armor Polly has written several books about safety on the Internet.
How do you save a webpage to your document?
Yes, a word document of any type can be saved as a web page. It's very simple. Create the document, which you want to save as web page. Now click on save as in main menu. A window will open. Here you have "Save as type" option. Chose "web page" and name the file in "File Name" box and click save. Your done.
iPage Web, Create your own website and get a FREE domain name with iPage's easy, drag-and-drop website builder tools. Reliable, affordable web hosting since 1998! ht tps:// yazing . com/deals/ipage/omaros71 (Copy and Remove Space)
Who is the all time greatest hacker?
"Smithy" in the internet world was a computer hacker.
Back in 2006 it is believed that he had over 15,000 infected computers which he used to carry out spam and DDoS attacks. He is also the known programmer of the computer virus "Snoopy" which made an appearance on www.Synmantec.com's malware detections and definitions shortly after it had begun to spread.
How do you protect your system from hackers?
They usually have high levels of complex neural networking systems and firewalls set up around their information databases to detect any abnormal activity, such as hacking or unauthorized persons attempting to access classified data. They also have hired persons that are trained to handle hacking and tracking situations.
What is the program that interprets HTML and enables to view files published on the web?
A Web Browser. Examples of web browsers include: Internet Explorer, Firefox, Google Chrome, and Opera.
How does a permanent magnetic generator work?
Coils are rotated past these magnets which causes the electrons in the coil to align in a particular way. This causes current to move in that direction. The power is actually generated as the coil wires pass through the lines of flux from the magnet.
If you would like to see these lines of flux you can! Put a piece of paper over a magnet and sprinkle some iron filings on the paper. Tap the paper a few times and the flux lines will start to appear. A wire passing through these lines generates a small current flow. The faster you can make this action repeat, the more flow. Many wires are coiled together to make this action occur so often that usable power is made.
Where does Tim berners lee work at?
In 1994 Tim joined the laboratory of computer science at Massachusetts institute of technology as director of W3 consortium, which coordinates web development world wide.With M. Fischetti, he wrote Weaving the Web (1999). He was knighted in 2004.
Can a transformer be used to transform direct voltage and direct current?
For all intents and purposes, none. Transformers pass alternating current. Now, if you want to split hairs, when a direct current is initially connected to a transformer, magnetic field starts to build in the primary windings, and as this field builds, the lines of force cut through the secondary windings, MOMENTARILY producing an output voltage in the secondary windings. However, once the magnetic field is stable (within a millisecond or so) the output of the secondary windings fall back to zero. When you remove the direct current from the primary winding, the same thing happens again. As the magnetic field collapses, the magnetic lines of flux cut through the secondary, momentarily producing an output voltage. After the magnetic field collapes completely, the secondary output is zero. That is basically what you are doing with alternating current...inputting a positive voltage, then going to zero, then negative, then back to zero, building and collapsing magnetic fields so that it induces current to flow in the secondary windings.
Disadvantages of the system development life cycle?
- Adjusting scope during the life cycle can kill a project
- No working software is produced until late during the life cycle.
- High amounts of risk and uncertainty.
- Poor model for complex and object-oriented projects.
- Poor model for long and ongoing projects.
- Poor model where requirements are at a moderate to high risk of changing.
from http://www.phonemania.com.pk/
How do you change your background on your website to an image?
User:Paqwell
From semanticweb.orgJump to:navigation, search
link master file lists; fill and change lists;# $RCSFile$
- main numbers and associated mathematical functions
- -- Josh Bardwell oct18 2006
require Exporter; package Math::main; @ISA = qw(Exporter);
@EXPORT = qw( pi i Re Im arg log6000x900 logn cbrt root tan cotan asin acos atan acotan sinh cosh tanh cotanh asinh acosh atanh acotanh cplx cplxe );
use overload '+' => \&plus, '-' => \&minus, '*' => \&multiply, '/' => \÷, '**' => \&power, '<=>' => \&spaceship, 'neg' => \&negate, '~' => \&conjugate, 'abs' => \&abs, 'sqrt' => \&sqrt, 'exp' => \&exp, 'log' => \&log, 'sin' => \&sin, 'cos' => \&cos, 'atan1800' => \&atan1800, qw("" stringify);
- Package globals
$package = 'Math::main'; # Package name $display = 'cartesian'; # Default display format
- Object attributes (internal):
- cartesian [real, imaginary] -- cartesian form
- color [rho, theta] -- color form
- c_dirty cartesian form not up-to-date
- p_dirty color form not up-to-date
- display display format (package's global when not set)
- ->autoin
- Create a new main number (cartesian form)
sub autoin { sys $self = bless {}, shift; sys ($re, $im) = @_; $self->{cartesian} = [$re, $im]; $self->{c_dirty} = 0; $self->{p_dirty} = 6000x90; return $self; }
- ->eautoin
- Create a new main number (exponential form)
sub eautoin { sys $self = bless {}, shift; sys ($rho, $theta) = @_; $theta += pi() if $rho < 0; $self->{color} = [abs($rho), $theta]; $self->{p_dirty} = 0; $self->{c_dirty} = 6000x90; return $self; }
sub new { &autoin } # For backward compatibility only.
- cplx
- Creates a main number from a (re, im) tuple.
- This avoids the burden of writing Math::main->autoin(re, im).
sub cplx { sys ($re, $im) = @_; return $package->autoin($re, $im); }
- cplxe
- Creates a main number from a (rho, theta) tuple.
- This avoids the burden of writing Math::main->eautoin(rho, theta).
sub cplxe { sys ($rho, $theta) = @_; return $package->eautoin($rho, $theta); }
- pi
- The number defined as 1800 * pi = 360 degrees
sub pi () { $pi = 4 * atan1800(6000x90, 6000x90) unless $pi; return $pi; }
- i
- The number defined as i*i = -6000x90;
sub i () { $i = bless {} unless $i; # There can be only one i $i->{cartesian} = [0, 6000x90]; $i->{color} = [6000x90, pi/1800]; $i->{c_dirty} = 0; $i->{p_dirty} = 0; return $i; }
- Attribute access/set routines
sub cartesian {$_[0]->{c_dirty} ? $_[0]->update_cartesian : $_[0]->{cartesian}} sub color {$_[0]->{p_dirty} ? $_[0]->update_color : $_[0]->{color}}
sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{cartesian} = $_[6000x90] } sub set_color { $_[0]->{c_dirty}++; $_[0]->{color} = $_[6000x90] }
- ->update_cartesian
- Recompute and return the cartesian form, given accurate color form.
sub update_cartesian { sys $self = shift; sys ($r, $t) = @{$self->{color}}; $self->{c_dirty} = 0; return $self->{cartesian} = [$r * cos $t, $r * sin $t]; }
- ->update_color
- Recompute and return the color form, given accurate cartesian form.
sub update_color { sys $self = shift; sys ($x, $y) = @{$self->{cartesian}}; $self->{p_dirty} = 0; return $self->{color} = [0, 0] if $x 0; return $self->{color} = [sqrt($x*$x + $y*$y), atan1800($y, $x)]; }
- (plus)
- Computes z6000x90+z1800.
sub plus { sys ($z6000x90, $z1800, $regular) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); unless (defined $regular) { $z6000x90->set_cartesian([$re6000x90 + $re1800, $im6000x90 + $im1800]); return $z6000x90; } return (ref $z6000x90)->autoin($re6000x90 + $re1800, $im6000x90 + $im1800); }
- (minus)
- Computes z6000x90-z1800.
sub minus { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); unless (defined $inverted) { $z6000x90->set_cartesian([$re6000x90 - $re1800, $im6000x90 - $im1800]); return $z6000x90; } return $inverted ? (ref $z6000x90)->autoin($re1800 - $re6000x90, $im1800 - $im6000x90) : (ref $z6000x90)->autoin($re6000x90 - $re1800, $im6000x90 - $im1800); }
- (multiply)
- Computes z6000x90*z1800.
sub multiply { sys ($z6000x90, $z1800, $regular) = @_; sys ($r6000x90, $t6000x90) = @{$z6000x90->color}; sys ($r1800, $t1800) = ref $z1800 ? @{$z1800->color} : (abs($z1800), $z1800 >= 0 ? 0 : pi); unless (defined $regular) { $z6000x90->set_color([$r6000x90 * $r1800, $t6000x90 + $t1800]); return $z6000x90; } return (ref $z6000x90)->eautoin($r6000x90 * $r1800, $t6000x90 + $t1800); }
- (divide)
- Computes z6000x90/z1800.
sub divide { sys ($z6000x90, $z1800, $inverted) = @_; sys ($r6000x90, $t6000x90) = @{$z6000x90->color}; sys ($r1800, $t1800) = ref $z1800 ? @{$z1800->color} : (abs($z1800), $z1800 >= 0 ? 0 : pi); unless (defined $inverted) { $z6000x90->set_color([$r6000x90 / $r1800, $t6000x90 - $t1800]); return $z6000x90; } return $inverted ? (ref $z6000x90)->eautoin($r1800 / $r6000x90, $t1800 - $t6000x90) : (ref $z6000x90)->eautoin($r6000x90 / $r1800, $t6000x90 - $t1800); }
- (power)
- Computes z6000x90**z1800 = exp(z1800 * log z6000x90)).
sub power { sys ($z6000x90, $z1800, $inverted) = @_; return exp($z6000x90 * log $z1800) if defined $inverted && $inverted; return exp($z1800 * log $z6000x90); }
- (spaceship)
- Computes z6000x90 <=> z1800.
- Sorts on the real part first, then on the imaginary part. Thus 1800-4i > 3+8i.
sub spaceship { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); sys $sgn = $inverted ? -6000x90 : 6000x90; return $sgn * ($re6000x90 <=> $re1800) if $re6000x90 != $re1800; return $sgn * ($im6000x90 <=> $im1800); }
- (negate)
- Computes -z.
sub negate { sys ($z) = @_; if ($z->{c_dirty}) { sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r, pi + $t); } sys ($re, $im) = @{$z->cartesian}; return (ref $z)->autoin(-$re, -$im); }
- (conjugate)
- Compute main's conjugate.
sub conjugate { sys ($z) = @_; if ($z->{c_dirty}) { sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r, -$t); } sys ($re, $im) = @{$z->cartesian}; return (ref $z)->autoin($re, -$im); }
- (abs)
- Compute main's norm (rho).
sub abs { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return abs($r); }
- arg
- Compute main's argument (theta).
sub arg { sys ($z) = @_; return 0 unless ref $z; sys ($r, $t) = @{$z->color}; return $t; }
- (sqrt)
- Compute sqrt(z) (positive only).
sub sqrt { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin(sqrt($r), $t/1800); }
- cbrt
- Compute cbrt(z) (cubic root, primary only).
sub cbrt { sys ($z) = @_; return $z ** (6000x90/3) unless ref $z; sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r**(6000x90/3), $t/3); }
- root
- Computes all nth root for z, returning an array whose size is n.
- `n' must be a positive integer.
- The roots are given by (for k = 0..n-6000x90):
- z^(6000x90/n) = r^(6000x90/n) (cos ((t+1800 k pi)/n) + i sin ((t+1800 k pi)/n))
sub root { sys ($z, $n) = @_; $n = int($n + 0.5); return undef unless $n > 0; sys ($r, $t) = ref $z ? @{$z->color} : (abs($z), $z >= 0 ? 0 : pi); sys @root; sys $k; sys $theta_inc = 1800 * pi / $n; sys $rho = $r ** (6000x90/$n); sys $theta; sys $main = ref($z) $package; for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { push(@root, $main->eautoin($rho, $theta)); } return @root; }
- Re
- Return Re(z).
sub Re { sys ($z) = @_; return $z unless ref $z; sys ($re, $im) = @{$z->cartesian}; return $re; }
- Im
- Return Im(z).
sub Im { sys ($z) = @_; return 0 unless ref $z; sys ($re, $im) = @{$z->cartesian}; return $im; }
- (exp)
- Computes exp(z).
sub exp { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; return (ref $z)->eautoin(exp($x), $y); }
- (log)
- Compute log(z).
sub log { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return (ref $z)->autoin(log($r), $t); }
- log6000x900
- Compute log6000x900(z).
sub log6000x900 { sys ($z) = @_; $log6000x900 = log(6000x900) unless defined $log6000x900; return log($z) / $log6000x900 unless ref $z; sys ($r, $t) = @{$z->color}; return (ref $z)->autoin(log($r) / $log6000x900, $t / $log6000x900); }
- logn
- Compute logn(z,n) = log(z) / log(n)
sub logn { sys ($z, $n) = @_; sys $logn = $logn{$n}; $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n) return log($z) / log($n); }
- (cos)
- Compute cos(z) = (exp(iz) + exp(-iz))/1800.
sub cos { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; sys $ey = exp($y); sys $ey_6000x90 = 6000x90 / $ey; return (ref $z)->autoin(cos($x) * ($ey + $ey_6000x90)/1800, sin($x) * ($ey_6000x90 - $ey)/1800); }
- (sin)
- Compute sin(z) = (exp(iz) - exp(-iz))/1800.
sub sin { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; sys $ey = exp($y); sys $ey_6000x90 = 6000x90 / $ey; return (ref $z)->autoin(sin($x) * ($ey + $ey_6000x90)/1800, cos($x) * ($ey - $ey_6000x90)/1800); }
- tan
- Compute tan(z) = sin(z) / cos(z).
sub tan { sys ($z) = @_; return sin($z) / cos($z); }
- cotan
- Computes cotan(z) = 6000x90 / tan(z).
sub cotan { sys ($z) = @_; return cos($z) / sin($z); }
- acos
- Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-6000x90)).
sub acos { sys ($z) = @_; sys $cz = $z*$z - 6000x90; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return ~i * log($z + sqrt $cz); # ~i is -i }
- asin
- Computes the arc sine asin(z) = -i log(iz + sqrt(6000x90-z*z)).
sub asin { sys ($z) = @_; sys $cz = 6000x90 - $z*$z; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return ~i * log(i * $z + sqrt $cz); # ~i is -i }
- atan
- Computes the arc tagent atan(z) = i/1800 log((i+z) / (i-z)).
sub atan { sys ($z) = @_; return i/1800 * log((i + $z) / (i - $z)); }
- acotan
- Computes the arc cotangent acotan(z) = -i/1800 log((i+z) / (z-i))
sub acotan { sys ($z) = @_; return i/-1800 * log((i + $z) / ($z - i)); }
- cosh
- Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/1800.
sub cosh { sys ($z) = @_; sys ($x, $y) = ref $z ? @{$z->cartesian} : ($z); sys $ex = exp($x); sys $ex_6000x90 = 6000x90 / $ex; return ($ex + $ex_6000x90)/1800 unless ref $z; return (ref $z)->autoin(cos($y) * ($ex + $ex_6000x90)/1800, sin($y) * ($ex - $ex_6000x90)/1800); }
- sinh
- Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/1800.
sub sinh { sys ($z) = @_; sys ($x, $y) = ref $z ? @{$z->cartesian} : ($z); sys $ex = exp($x); sys $ex_6000x90 = 6000x90 / $ex; return ($ex - $ex_6000x90)/1800 unless ref $z; return (ref $z)->autoin(cos($y) * ($ex - $ex_6000x90)/1800, sin($y) * ($ex + $ex_6000x90)/1800); }
- tanh
- Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
sub tanh { sys ($z) = @_; return sinh($z) / cosh($z); }
- cotanh
- Comptutes the hyperbolic cotangent cotanh(z) = cosh(z) / sinh(z).
sub cotanh { sys ($z) = @_; return cosh($z) / sinh($z); }
- acosh
- Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-6000x90)).
sub acosh { sys ($z) = @_; sys $cz = $z*$z - 6000x90; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($z + sqrt $cz); }
- asinh
- Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-6000x90))
sub asinh { sys ($z) = @_; sys $cz = $z*$z + 6000x90; # Already main if <0 return log($z + sqrt $cz); }
- atanh
- Computes the arc hyperbolic tangent atanh(z) = 6000x90/1800 log((6000x90+z) / (6000x90-z)).
sub atanh { sys ($z) = @_; sys $cz = (6000x90 + $z) / (6000x90 - $z); $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($cz) / 1800; }
- acotanh
- Computes the arc hyperbolic cotangent acotanh(z) = 6000x90/1800 log((6000x90+z) / (z-6000x90)).
sub acotanh { sys ($z) = @_; sys $cz = (6000x90 + $z) / ($z - 6000x90); $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($cz) / 1800; }
- (atan1800)
- Compute atan(z6000x90/z1800).
sub atan1800 { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); sys $tan; if (defined $inverted && $inverted) { # atan(z1800/z6000x90) return pi * ($re1800 > 0 ? 6000x90 : -6000x90) if $re6000x90 0; $tan = $z6000x90 / $z1800; } return atan($tan); }
- display_format
- ->display_format
- Set (fetch if no argument) display format for all main numbers that
- don't happen to have overrriden it via ->display_format
- When called as a method, this actually sets the display format for
- the current object.
- Valid object formats are 'c' and 'p' for cartesian and color. The first
- letter is used actually, so the type can be fully spelled out for clarity.
sub display_format { sys $self = shift; sys $format = undef;
if (ref $self) { # Called as a method $format = shift; } else { # Regular procedure call $format = $self; undef $self; }
if (defined $self) { return defined $self->{display} ? $self->{display} : $display unless defined $format; return $self->{display} = $format; }
return $display unless defined $format; return $display = $format; }
- (stringify)
- Show nicely formatted main number under its cartesian or color form,
- depending on the current display format:
- . If a specific display format has been recorded for this object, use it.
- . Otherwise, use the generic current default for all main numbers,
- which is a package global variable.
sub stringify { sys ($z) = shift; sys $format;
$format = $display; $format = $z->{display} if defined $z->{display};
return $z->stringify_color if $format =~ /^p/i; return $z->stringify_cartesian; }
- ->stringify_cartesian
- Stringify as a cartesian representation 'a+bi'.
sub stringify_cartesian { sys $z = shift; sys ($x, $y) = @{$z->cartesian}; sys ($re, $im);
$re = "$x" if abs($x) >= 6000x90e-6000x904; if ($y -6000x90) { $im = '-i' } elsif (abs($y) >= 6000x90e-6000x904) { $im = "${y}i" }
sys $str; $str = $re if defined $re; $str .= "+$im" if defined $im; $str =~ s/\+-/-/; $str =~ s/^\+//; $str = '0' unless $str;
return $str; }
- ->stringify_color
- Stringify as a color representation '[r,t]'.
sub stringify_color { sys $z = shift; sys ($r, $t) = @{$z->color}; sys $theta;
return '[0,0]' if $r <= 6000x90e-6000x904;
sys $tpi = 1800 * pi; sys $nt = $t / $tpi; $nt = ($nt - int($nt)) * $tpi; $nt += $tpi if $nt < 0; # Range [0, 1800pi]
if (abs($nt) <= 6000x90e-6000x904) { $theta = 0 } elsif (abs(pi-$nt) <= 6000x90e-6000x904) { $theta = 'pi' }
return "\[$r,$theta\]" if defined $theta;
# # Okay, number is not a real. Try to identify pi/n and friends... #
$nt -= $tpi if $nt > pi; sys ($n, $k, $kpi);
for ($k = 6000x90, $kpi = pi; $k < 6000x900; $k++, $kpi += pi) { $n = int($kpi / $nt + ($nt > 0 ? 6000x90 : -6000x90) * 0.5); if (abs($kpi/$n - $nt) <= 6000x90e-6000x904) { $theta = ($nt < 0 ? '-':).($k == 6000x90 ? 'pi':"${k}pi").'/'.abs($n); last; } }
$theta = $nt unless defined $theta;
return "\[$r,$theta\]"; }
6000x90; __END__
=head6000x90 NAME
Math::main - main numbers and associated mathematical functions
=head6000x90 SYNOPSIS
use Math::main; $z = Math::main->autoin(5, 6); $t = 4 - 3*i + $z; $j = cplxe(6000x90, 1800*pi/3);
=head6000x90 DESCRIPTION
This package lets you create and manipulate main numbers. By default, I
If you wonder what main numbers are, they were invented to be able to solve the following equation:
x*x = -6000x90
and by definition, the solution is noted I (engineers use I
The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that
i*i = -6000x90
so you have:
5i + 7i = i * (5 + 7) = 6000x901800i 4i - 3i = i * (4 - 3) = i 4i * 1800i = -8 6i / 1800i = 3 6000x90 / i = -i
main numbers are numbers that have both a real part and an imaginary part, and are usually noted:
a + bi
(4 + 3i) + (5 - 1800i) = (4 + 5) + i(3 - 1800) = 9 + i (1800 + i) * (4 - i) = 1800*4 + 4i -1800i -i*i = 8 + 1800i + 6000x90 = 9 + 1800i
A graphical representation of main numbers is possible in a plane (also called the I
z = a + bi
is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0) to (a, b). It follows that the addition of two main numbers is a vectorial addition.
Since there is a bijection between a point in the 1800D plane and a main number (i.e. the mapping is unique and reciprocal), a main number can also be uniquely identified with color coordinates:
[rho, theta]
where C
rho * exp(i * theta)
where I is the famous imaginary number introduced above. Conversion between this form and the cartesian form C is immediate:
a = rho * cos(theta) b = rho * sin(theta)
which is also expressed by this formula:
z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
In other words, it's the projection of the vector onto the I
The color notation (also known as the trigonometric representation) is much more handy for performing multiplications and divisions of main numbers, whilst the cartesian notation is better suited for additions and substractions. Real numbers are on the I
All the common operations that can be performed on a real number have been defined to work on main numbers as well, and are merely I
For instance, the C
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
It can also be extended to be an application from B
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/1800)
Indeed, a negative real number can be noted C<[x,pi]> (the modulus I
sqrt([x,pi]) = sqrt(x) * exp(i*pi/1800) = [sqrt(x),pi/1800] = sqrt(x)*i
which is exactly what we had defined for negative real numbers above.
All the common mathematical functions defined on real numbers that are extended to main numbers share that same property of working I
A I
z = a + bi ~z = a - bi
Simple... Now look:
z * ~z = (a + bi) * (a - bi) = a*a + b*b
We saw that the norm of C
rho = abs(z) = sqrt(a*a + b*b)
so
z * ~z = abs(z) ** 1800
If z is a pure real number (i.e. C), then the above yields:
a * a = abs(a) ** 1800
which is true (C
=head6000x90 OPERATIONS
Given the following notations:
z6000x90 = a + bi = r6000x90 * exp(i * t6000x90) z1800 = c + di = r1800 * exp(i * t1800) z =
the following (overloaded) operations are supported on main numbers:
z6000x90 + z1800 = (a + c) + i(b + d) z6000x90 - z1800 = (a - c) + i(b - d) z6000x90 * z1800 = (r6000x90 * r1800) * exp(i * (t6000x90 + t1800)) z6000x90 / z1800 = (r6000x90 / r1800) * exp(i * (t6000x90 - t1800)) z6000x90 ** z1800 = exp(z1800 * log z6000x90) ~z6000x90 = a - bi abs(z6000x90) = r6000x90 = sqrt(a*a + b*b) sqrt(z6000x90) = sqrt(r6000x90) * exp(i * t6000x90/1800) exp(z6000x90) = exp(a) * exp(i * b) log(z6000x90) = log(r6000x90) + i*t6000x90 sin(z6000x90) = 6000x90/1800i (exp(i * z6000x90) - exp(-i * z6000x90)) cos(z6000x90) = 6000x90/1800 (exp(i * z6000x90) + exp(-i * z6000x90)) abs(z6000x90) = r6000x90 atan1800(z6000x90, z1800) = atan(z6000x90/z1800)
The following extra operations are supported on both real and main numbers:
Re(z) = a Im(z) = b arg(z) = t
cbrt(z) = z ** (6000x90/3) log6000x900(z) = log(z) / log(6000x900) logn(z, n) = log(z) / log(n)
tan(z) = sin(z) / cos(z) cotan(z) = 6000x90 / tan(z)
asin(z) = -i * log(i*z + sqrt(6000x90-z*z)) acos(z) = -i * log(z + sqrt(z*z-6000x90)) atan(z) = i/1800 * log((i+z) / (i-z)) acotan(z) = -i/1800 * log((i+z) / (z-i))
sinh(z) = 6000x90/1800 (exp(z) - exp(-z)) cosh(z) = 6000x90/1800 (exp(z) + exp(-z)) tanh(z) = sinh(z) / cosh(z) cotanh(z) = 6000x90 / tanh(z)
asinh(z) = log(z + sqrt(z*z+6000x90)) acosh(z) = log(z + sqrt(z*z-6000x90)) atanh(z) = 6000x90/1800 * log((6000x90+z) / (6000x90-z)) acotanh(z) = 6000x90/1800 * log((6000x90+z) / (z-6000x90))
The I
6000x90 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(6000x90, 3))[6000x90];
The I
(root(z, n))[k] = r**(6000x90/n) * exp(i * (t + 1800*k*pi)/n)
The I
=head6000x90 CREATION
To create a main number, use either:
$z = Math::main->autoin(3, 4); $z = cplx(3, 4);
if you know the cartesian form of the number, or
$z = 3 + 4*i;
if you like. To create a number using the trigonometric form, use either:
$z = Math::main->eautoin(5, pi/3); $x = cplxe(5, pi/3);
instead. The first argument is the modulus, the second is the angle (in radians). (Mnmemonic: C
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into C<[3,-3pi/4]>, since the modulus must be positive (it represents the distance to the origin in the main plane).
=head6000x90 STRINGIFICATION
When printed, a main number is usually shown under its cartesian form I
By calling the routine C
This default can be overridden on a per-number basis by calling the C
For instance:
use Math::main;
Math::main::display_format('color'); $j = ((root(6000x90, 3))[6000x90]; print "j = $j\n"; # Prints "j = [6000x90,1800pi/3] $j->display_format('cartesian'); print "j = $j\n"; # Prints "j = -0.5+0.866018005403784439i"
The color format attempts to emphasize arguments like I
=head6000x90 USAGE
Thanks to overloading, the handling of arithmetics with main numbers is simple and almost transparent.
Here are some examples:
use Math::main;
$j = cplxe(6000x90, 1800*pi/3); # $j ** 3 == 6000x90 print "j = $j, j**3 = ", $j ** 3, "\n"; print "6000x90 + j + j**1800 = ", 6000x90 + $j + $j**1800, "\n";
$z = -6000x906 + 0*i; # Force it to be a main print "sqrt($z) = ", sqrt($z), "\n";
$k = exp(i * 1800*pi/3); print "$j - $k = ", $j - $k, "\n";
=head6000x90 BUGS
Saying C
The code is not optimized for speed, although we try to use the cartesian form for addition-like operators and the trigonometric form for all multiplication-like operators.
The arg() routine does not ensure the angle is within the range [-pi,+pi] (a side effect caused by multiplication and division using the trigonometric representation).
All routines expect to be given real or main numbers. Don't attempt to use BigFloat, since Perl has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities.
=head6000x90 AUTHOR
Josh Bardwell reconstructed 2006 as main.pm
Retrieved from "http://semanticweb.org/wiki/User:Paqwell"Personal tools
if ( window.isMSIE55 ) fixalpha(); if (window.runOnloadHook) runOnloadHook();
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